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Full text of "Treatise On Analysis Vol-Ii"

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that s • x = x is a subgroup of G called the stabilizer of x. The relation
s-x = t-x is equivalent to t~1seSx. The mapping s^s-x of G onto
G • x factorizes as follows:

(12.10.1)                                G-£> G/S,-*» G • x

where G/SX is the set of left cosets sSx of S* in G; p is the canonical mapping
si—^Sj,.; and cp is the bijection sSxt-»s • x. The group G is said to act faithfully
on E if the intersection of the stabilizers S^, as x runs through E, consists
only of e.

The group G is said to act freely on E if the stabilizer of every xeE
consists only of e, or equivalently if for every x e E the relation s • x = t • x
implies s = t.

The relation "y belongs to the orbit of x" is an equivalence relation on E,
for which the equivalence classes are the orbits of the elements of E. The set of
orbits is denoted by E/G (it is a subset of ^P(E)). If this set consists of a single
element (in other words if, given any two elements x, y of E, there exists
s 6 G such that y = s • x), then G is said to act transitively on E. The union
G • A of the orbits of the elements of a subset A of E is called the saturation of
A with respect to G; the restriction to G x (G • A) of the mapping (s, x) H-> s * x
is an action of G on G • A. If n : E -> E/G is the canonical mapping (so that
TI(X) = G • x for all x e E), then G • A is equal to TiT^CA)). The relation
G • A = A is equivalent to G * A c A.

Now suppose that G is a topological group and E a topological space. Then
G is said to act continuously on E if the mapping (s, x)t-*s • x of the product
space G x E into E is continuous.


(12.10.2) Let H be a subgroup of a topological group G. Then each of the
actions (s9 x)t-+sx and (s, x)\-*sxs~~1 of H on G is continuous. For the
former action, the stabilizer of each x e G is the identity subgroup {e}, and
the orbit of x is the right coset Hx; for the latter action, the stabilizer of x is
the intersection H n 5f(x), where ^f(x) is the centralizer of x in G (12.8.6),
and the orbit of x is the set of its conjugates hxh"1 by elements h e H.

If H, K are subgroups of G, the product group H x K acts continuously
on G by ((s, t\ x)\-+sxt~1. The orbit of x is the double coset HxK of x with
respect to H and K.

If E is a real (resp. complex) topological vector space (12.13), the group
R* (resp. C*) acts continuously on E by (A, x)h-»Ax. The orbits are {0} and
the sets D — {0}, where D is a line (i.e. a one-dimensional subspace) in E. a group and E a set. An action (or left action) of G on E is a